Let $G$ be a simple graph‎. ‎An independent set is a set of‎ ‎pairwise non-adjacent vertices‎. ‎The number of vertices in a maximum independent set of $G$ is‎ ‎denoted by $\alpha(G)$‎. ‎In this paper‎, ‎we characterize graphs $G$ with $n$ vertices and with maximum‎ ‎number of maximum independent sets provided that $\alpha(G)\leq 2$ or $\alpha(G)\geq n-3$‎.